Theorem serzsplit 7056

Description: Split off an initial piece of the partial sum of an infinite series.

Hypothesis

Ref	Expression
serzsplit.1	|- F e. V

Assertion

Ref	Expression
serzsplit	|- ((N e. ZZ /\ K e. (M...(N - 1)) /\ A.k e. (M...N)(F` k) e. CC) -> ((<.M, + >. seq F)` N) = (((<.M, + >. seq F)` K) + ((<.(K + 1), + >. seq F)` N)))

Distinct variable groups:   k,F   k,M   k,N

Proof of Theorem serzsplit

Step	Hyp	Ref	Expression
1		fsumsplit 7020	. 2 |- ((N e. ZZ /\ K e. (M...(N - 1)) /\ A.k e. (M...N)(F` k) e. CC) -> sum_k e. (M...N)(F` k) = (sum_k e. (M...K)(F` k) + sum_k e. ((K + 1)...N)(F` k)))
2		zcnt 6142	. . . . . . 7 |- (N e. ZZ -> N e. CC)
3		ax1cn 5281	. . . . . . . 8 |- 1 e. CC
4		npcant 5411	. . . . . . . 8 |- ((N e. CC /\ 1 e. CC) -> ((N - 1) + 1) = N)
5	3, 4	mpan2 698	. . . . . . 7 |- (N e. CC -> ((N - 1) + 1) = N)
6	2, 5	syl 10	. . . . . 6 |- (N e. ZZ -> ((N - 1) + 1) = N)
7	6	adantr 391	. . . . 5 |- ((N e. ZZ /\ K e. (M...(N - 1))) -> ((N - 1) + 1) = N)
8		oprex 3989	. . . . . . . 8 |- (N - 1) e. V
9		elfzuz2t 6487	. . . . . . . 8 |- (((N - 1) e. V /\ K e. (M...(N - 1))) -> (N - 1) e. (ZZ>` M))
10	8, 9	mpan 697	. . . . . . 7 |- (K e. (M...(N - 1)) -> (N - 1) e. (ZZ>` M))
11		peano2uz 6448	. . . . . . 7 |- ((N - 1) e. (ZZ>` M) -> ((N - 1) + 1) e. (ZZ>` M))
12	10, 11	syl 10	. . . . . 6 |- (K e. (M...(N - 1)) -> ((N - 1) + 1) e. (ZZ>` M))
13	12	adantl 390	. . . . 5 |- ((N e. ZZ /\ K e. (M...(N - 1))) -> ((N - 1) + 1) e. (ZZ>` M))
14	7, 13	eqeltrrd 1552	. . . 4 |- ((N e. ZZ /\ K e. (M...(N - 1))) -> N e. (ZZ>` M))
15		serzsplit.1	. . . . 5 |- F e. V
16	15	fsumserz 6999	. . . 4 |- (N e. (ZZ>` M) -> sum_k e. (M...N)(F` k) = ((<.M, + >. seq F)` N))
17	14, 16	syl 10	. . 3 |- ((N e. ZZ /\ K e. (M...(N - 1))) -> sum_k e. (M...N)(F` k) = ((<.M, + >. seq F)` N))
18	17	3adant3 801	. 2 |- ((N e. ZZ /\ K e. (M...(N - 1)) /\ A.k e. (M...N)(F` k) e. CC) -> sum_k e. (M...N)(F` k) = ((<.M, + >. seq F)` N))
19		elfzuzt 6489	. . . . . 6 |- (K e. (M...(N - 1)) -> K e. (ZZ>` M))
20	19	adantl 390	. . . . 5 |- ((N e. ZZ /\ K e. (M...(N - 1))) -> K e. (ZZ>` M))
21	15	fsumserz 6999	. . . . 5 |- (K e. (ZZ>` M) -> sum_k e. (M...K)(F` k) = ((<.M, + >. seq F)` K))
22	20, 21	syl 10	. . . 4 |- ((N e. ZZ /\ K e. (M...(N - 1))) -> sum_k e. (M...K)(F` k) = ((<.M, + >. seq F)` K))
23		elfzuz3t 6479	. . . . . . . . 9 |- (((N - 1) e. V /\ K e. (M...(N - 1))) -> (N - 1) e. (ZZ>` K))
24	8, 23	mpan 697	. . . . . . . 8 |- (K e. (M...(N - 1)) -> (N - 1) e. (ZZ>` K))
25		eluzp1p1t 6436	. . . . . . . 8 |- ((N - 1) e. (ZZ>` K) -> ((N - 1) + 1) e. (ZZ>` (K + 1)))
26	24, 25	syl 10	. . . . . . 7 |- (K e. (M...(N - 1)) -> ((N - 1) + 1) e. (ZZ>` (K + 1)))
27	26	adantl 390	. . . . . 6 |- ((N e. ZZ /\ K e. (M...(N - 1))) -> ((N - 1) + 1) e. (ZZ>` (K + 1)))
28	7, 27	eqeltrrd 1552	. . . . 5 |- ((N e. ZZ /\ K e. (M...(N - 1))) -> N e. (ZZ>` (K + 1)))
29	15	fsumserz 6999	. . . . 5 |- (N e. (ZZ>` (K + 1)) -> sum_k e. ((K + 1)...N)(F` k) = ((<.(K + 1), + >. seq F)` N))
30	28, 29	syl 10	. . . 4 |- ((N e. ZZ /\ K e. (M...(N - 1))) -> sum_k e. ((K + 1)...N)(F` k) = ((<.(K + 1), + >. seq F)` N))
31	22, 30	opreq12d 3984	. . 3 |- ((N e. ZZ /\ K e. (M...(N - 1))) -> (sum_k e. (M...K)(F` k) + sum_k e. ((K + 1)...N)(F` k)) = (((<.M, + >. seq F)` K) + ((<.(K + 1), + >. seq F)` N)))
32	31	3adant3 801	. 2 |- ((N e. ZZ /\ K e. (M...(N - 1)) /\ A.k e. (M...N)(F` k) e. CC) -> (sum_k e. (M...K)(F` k) + sum_k e. ((K + 1)...N)(F` k)) = (((<.M, + >. seq F)` K) + ((<.(K + 1), + >. seq F)` N)))
33	1, 18, 32	3eqtr3d 1518	1 |- ((N e. ZZ /\ K e. (M...(N - 1)) /\ A.k e. (M...N)(F` k) e. CC) -> ((<.M, + >. seq F)` N) = (((<.M, + >. seq F)` K) + ((<.(K + 1), + >. seq F)` N)))

Syntax hints:   -> wi 3   /\ wa 223   /\ w3a 777   = wceq 958   e. wcel 960  A.wral 1648  Vcvv 1814  <.cop 2415  ` cfv 3188  (class class class)co 3969  CCcc 5244  1c1 5247   + caddc 5249   - cmin 5304  ZZcz 5310  ZZ>cuz 6418  ...cfz 6468   seq cseqz 6532  sum_csu 6979

This theorem is referenced by:  clim2serzt 7134  clim2serz 7145

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-rep 2698  ax-sep 2708  ax-nul 2715  ax-pow 2748  ax-pr 2785  ax-un 2872  ax-inf2 4634

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 778  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-nel 1591  df-ral 1652  df-rex 1653  df-reu 1654  df-rab 1655  df-v 1815  df-sbc 1945  df-csb 2005  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-pss 2058  df-nul 2284  df-if 2366  df-pw 2406  df-sn 2416  df-pr 2417  df-tp 2419  df-op 2420  df-uni 2508  df-int 2538  df-iun 2572  df-br 2625  df-opab 2672  df-tr 2686  df-eprel 2838  df-id 2841  df-po 2846  df-so 2856  df-fr 2923  df-we 2940  df-ord 2957  df-on 2958  df-lim 2959  df-suc 2960  df-om 3138  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198  df-fn 3199  df-f 3200  df-f1 3201  df-fo 3202  df-f1o 3203  df-fv 3204  df-rdg 3938  df-opr 3971  df-oprab 3972  df-1st 4085  df-2nd 4086  df-1o 4139  df-oadd 4141  df-omul 4142  df-er 4267  df-ec 4269  df-qs 4272  df-en 4374  df-dom 4375  df-sdom 4376  df-ni 5012  df-pli 5013  df-mi 5014  df-lti 5015  df-plpq 5047  df-mpq 5048  df-enq 5049  df-nq 5050  df-plq 5051  df-mq 5052  df-rq 5053  df-ltq 5054  df-1q 5055  df-np 5098  df-1p 5099  df-plp 5100  df-mp 5101  df-ltp 5102  df-plpr 5176  df-mpr 5177  df-enr 5178  df-nr 5179  df-plr 5180  df-mr 5181  df-ltr 5182  df-0r 5183  df-1r 5184  df-m1r 5185  df-c 5252  df-0 5253  df-1 5254  df-i 5255  df-r 5256  df-plus 5257  df-mul 5258  df-lt 5259  df-sub 5368  df-neg 5370  df-pnf 5499  df-mnf 5500  df-xr 5501  df-ltxr 5502  df-le 5503  df-n 5927  df-n0 6102  df-z 6138  df-seq1 6309  df-shft 6342  df-uz 6419  df-fz 6469  df-seqz 6534  df-sum 6980

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